Basics of Fluid Mechanics, 2014a

10.1. INTRODUCTION 329
With the identity in (10.13) can be extend as
( ( ∂Ux
U ×∇×U = −i U y
∂y − ∂U y
∂x
( ( ∂Uy
− j U x
∂x − ∂U x
∂y
(
− k
) ( ∂Ux
+ U z
∂z − ∂U ) )
z
∂x
) ( ∂Uy
+ U z
∂z − ∂U ) )
z
∂y
( ∂Uz
U x
∂x − ∂U x
∂z
) ( ∂Uz
+ U y
∂y − ∂U ) )
y
(10.14)
∂z
The identity described in equation (10.14) is substituted into equation (10.12) to obtain
the form of
DU
Dt = ∂U
∂t + ∇ (U)2 − U ×∇×U (10.15)
Finally substituting equation (10.15) into the Euler equation to obtain a more convenient
form as
( )
∂U
ρ
∂t + ∇ (U)2 − U ×∇×U = −∇P − ∇ ρ g l (10.16)
A common assumption that employed in an isothermal flow is that density, ρ, is
a mere function of the static pressure, ρ = ρ(P ). According to this idea, the density is
constant when the pressure is constant. The mathematical interpretation of the pressure
gradient can be written as
∇P = dP ˆn (10.17)
dn
where ˆn is an unit vector normal to surface of constant property and the derivative
d /dn refers to the derivative in the direction of ˆn. Dividing equation (10.17) by the
density, ρ, yields
∇P
ρ = 1 dP
dn ρ ˆn = 1
dn
zero
net
effect
{}}{ ∫ ( ) dP
d
ρ
ˆn = d ∫ ( dP
dn ρ
)
∫ ( dP
ˆn = ∇
ρ
)
(10.18)
It can be noticed that taking a derivative after integration cancel both effects. The
derivative in the direction of ˆn is the gradient. This function is normal to the constant
of pressure, P , and therefore ∫ (dP / ρ) is function of the mere pressure.
Substituting equation (10.18) into equation (10.16) and collecting all terms under
the gradient yields
(
∂U
∂t + ∇ U 2 ∫ ( ) ) dP
2 + g l + = U ×∇×U (10.19)
ρ